Legislation currently requires health visitors to provide five health checks between pregnancy and when the child reaches two-and-a-half, although ministers are to consider axing this mandatory requirement as part of a controversial review launched this summer.
As part of a consultation on public health savings, Lewisham Council said the government review is "likely to give Lewisham and other local authorities more flexibility to target additional checks at the most vulnerable families".
It said that in the future, health visitors may only provide checks during pregnancy for women identified as vulnerable by maternity services.
"Other women will continue to have access to GPs and midwives for health checks during their pregnancy," the consultation document states.
Meanwhile, two of the mandatory checks - the seven- to 11-month check and the two- to 2.5-year check for families not identified as vulnerable - might be delivered in children's centres and in groups, rather than in the home.
It adds that the health visitors currently provide additional checks for some families at three to four months and three-and-a-half years, but may, in the future, only do this for families identified as "vulnerable".
Responsibility for the commissioning of public health services for children aged birth to five was shifted from NHS England to local authorities on 1 October 2015.
Lewisham Council said it needs to save money from its health visiting and school nursing budget because its public health grant will be reduced by £4.7m from April 2017 as a result of central government cuts.
The local authority has also suggested replacing school nurse drop-in sessions and one-to-one appointments with pupils, with a "teenage health service", which will offer online and face-to-face advice and support on health, wellbeing, substance misuse and sexual health.
"The school nursing service currently supports the health and emotional wellbeing of children and young people through school drop-ins, appointments and health promotion work. However, school nurses have limited capacity to do this work," the consultation document states.
The council has also proposed reducing school nurses, role in supporting children with disabilities and long-term conditions, and instead forming a "dedicated nursing team" supported by community children's doctors.
The council said it will also review the role of children's centres, with potential options including targeting services for families with higher needs, offering services alongside other existing health and education services, or integrating children's centres, one-to-one family support services with health visitor support.
A Lewisham Council spokeswoman said: "The government's decision to cut public health grants to councils by more than £160m within the next two years means some very difficult decisions will be made over the coming months.
"We want to protect the services most valued and needed by our communities, and at the same time achieve savings by delivering services in a different way, including savings of up to £2m from a new integrated health visiting and school nursing service. This is why we are continuing to consult with residents, service users and stakeholders."
The consultation closes on 14 August.